Our conventions for different kinds of indices, curvature, etc. follow Penrose and Rindler [1]}, [2]}.
The organization of this work is as follows.
We start in Section  with some background material: we review basic notions on
para-Hermitian geometry and related structures (keeping always in mind that we allow complex-valued maps),
generalized geometry and Lie and Courant algebroids, and conformal geometry and the Lee form.
Then in Section  we study almost para-Hermitian structures in 4-manifolds equipped with a metric
of different signatures, and we describe the spaces of such structures in each case.
In Section  we study integrability issues: half-integrability of para-Hermitian structures,
the Lie and Courant algebroids associated to special spacetimes,
and a possible notion of “algebroids” for the treatment of weighted fields.
In Section  we study some connections with twistor theory:
two- and three-dimensional twistor spaces, twistor families, and
integrability of small deformations of para-complex structures.
In Section  we make a summary of this work and comment on some possible future directions.
We include three appendices: Appendix  with a brief review of spinors in 4 dimensions
(in any signature), Appendix  with additional details about the
“gauge freedom” and the associated covariant formalism, and Appendix 
with some properties of the curvature of the natural connection in the above formalism.
